5.2. Exploring Quotients of Polynomial Functions. EXPLORE the Math. Each row shows the graphs of two polynomial functions.

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1 YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software Eploring Quotients of Polnomial Functions EXPLORE the Math Each row shows the graphs of two polnomial functions. A. GOAL Eplore graphs that are created b dividing polnomial functions. f() = g() = rational function a function that can be epressed as f() 5 p(), q() where p() and q() are polnomial functions, q() (e.g., f() 5 3,, and f() 5,, are rational functions, but f() 5 is #,, not because its denominator is not a polnomial) B. C. m() = p() = 5 n() = + 5 q() =? What are the characteristics of the graphs that are created b dividing two polnomial functions? A. Using the given functions, write the equation of the rational function 5 f (). Enter this equation into Y of the equation editor of a g() graphing calculator. Graph this equation using the window settings shown, and draw a sketch. 58 Eploring Quotients of Polnomial Functions

2 B. Describe the characteristics of the graph ou created in part A b answering the following questions: i) Where are the zeros? ii) Are there an asmptotes? If so, where are the? iii) What are the domain and range of this function? iv) Is it a continuous function? Eplain. v) Are there an values of 5 f () that are undefined? What g() feature(s) of the graph is (are) related to these values? vi) Describe the end behaviours of this function. vii) Is the resulting graph a function? Eplain. C. Write the equation defined b 5 g(). Predict how the graph of f () this function will differ from the graph of 5 f (). Graph this g() function using our graphing calculator, and draw a sketch. D. Describe the characteristics of the graph ou created in part C b answering the questions in part B. Tech Support When entering a rational function into a graphing calculator, use brackets around the epression in the numerator and the epression in the denominator. E. Repeat parts A through D for the functions in the other two rows. F. Using graphing technolog, and the same window settings ou used in part A, eplore the graphs of the following rational functions. Sketch each graph on separate aes, and note an holes or asmptotes. i) v) f () 5.5 f () 5 ii) vi) f () 5 f () 5 3 iii) iv) f () 5 3 f () 5 G. Eamine the graphs of the functions in parts i) and v) of part F at the point where. Eplain wh f () 5 5 has a hole where 5, but f () 5.5 has a vertical asmptote. Identif the other functions in part F that have holes and the other functions that have vertical asmptotes. vii) viii) f () 5 9 f () 5 3 Chapter 5 59

3 oblique asmptote an asmptote that is neither vertical nor horizontal, but slanted 3 3 H. Redraw the graph of the rational function f () 5.5. Then enter the equation into Y of the equation editor. What do ou notice? Eamine all our other sketches in this eploration to see if an of the other functions have an oblique asmptote. I. Eamine the equations with graphs that have horizontal asmptotes in part F. Compare the degree of the epression in the numerator with the degree of the epression in the denominator. Is there a connection between the degrees in the numerator and denominator and the eistence of horizontal asmptotes? Eplain. Repeat for functions with oblique asmptotes. J. Investigate several functions of the form f () 5 a b. Note c d similarities and differences. Without graphing, how can ou predict where a horizontal asmptote will occur? K. Investigate graphs of quotients of quadratic functions. How are the different from graphs of quotients of linear functions? L. Summarize the different characteristics of the graphs of rational functions. Reflecting M. How do the zeros of the function in the numerator help ou graph the rational function? How do the zeros of the function in the denominator help ou graph the rational function? N. Eplain how ou can use the epressions in the numerator and the denominator of a rational function to decide if the graph has i) a hole ii) a vertical asmptote iii) a horizontal asmptote iv) an oblique asmptote Eploring Quotients of Polnomial Functions

4 In Summar Ke Ideas The quotient of two polnomial functions results in a rational function which often has one or more discontinuities. The breaks or discontinuities in a rational function occur where the function is undefined. The function is undefined at values where the denominator is equal to zero. As a result, these values must be restricted from the domain of the function. The values that must be restricted from the domain of a rational function result in ke characteristics that define the shape of the graph. These characteristics include a combination of vertical asmptotes (also called infinite discontinuities) and holes (also called point discontinuities). The end behaviours of man rational functions are determined b either horizontal asmptotes or oblique asmptotes. Need to Know p(a) A rational function, f() 5 p(), has a hole at 5 a if. This q(a) 5 q() occurs when p() and q() contain a common factor of ( a). For eample, f() 5 has the common factor of ( ) in the numerator and the denominator. This results in a hole in the graph of f() at 5. f() = p(a) A rational function, f() 5 p(), has a vertical asmptote at 5 a if. q(a) 5 p(a) q() For eample, f() 5 has a vertical asmptote at 5. f() = + = A rational function, f() 5 p(), has a horizontal asmptote onl when the q() degree of p() is less than or equal to the degree of q(). For eample, f() 5 has a horizontal asmptote at 5. f() = + = = A rational function, f() 5 p(), has an oblique (slant) asmptote onl when q() the degree of p() is greater than the degree of q() b eactl. For eample, f() 5 has an oblique asmptote = 8 f() = + + = Chapter 5

5 FURTHER Your Understanding. Without using graphing technolog, match each equation with its corresponding graph. Eplain our reasoning. a) 5 d) 5 3 ( )( 3) b) 5 9 e) c) f) 5 5 ( 3) 3 A C E B D F. For each function, determine the equations of an vertical asmptotes, the locations of an holes, and the eistence of an horizontal or oblique asmptotes. a) 5 e) 5 i) 5 8 ( 3)( 5) b) 5 f ) 5 j) 5 3 c) 5 5 g) 5 3 k) d) 5 9 h) 5 l) Write an equation for a rational function with the properties as given. a) a hole at 5 b) a vertical asmptote anwhere and a horizontal asmptote along the -ais c) a hole at 5and a vertical asmptote at 5 d) a vertical asmptote at 5and a horizontal asmptote at 5 e) an oblique asmptote, but no vertical asmptote Eploring Quotients of Polnomial Functions